Resistance distance and Laplacian spectrum

Visualizing the Electrical Structure of Power Systems

“…Finally, note that resistance distances are closely related to random walks on graphs [69], and to the eigenvalues and eigenvectors of the graph's Laplacian, and they can be calculated by such means [70][71][72]. This is an example of spectral graph theory [54], whose application to power system problems has only recently emerged [35,61,[73][74][75][76].…”

Resistance Distance, Information Centrality, Node Vulnerability and Vibrations in Complex Networks

“…Such resistance is known to be a distance function [25] and called the resistance distance. It was introduced in a seminal paper by Klein and Randić a few years ago [25] and has been intensively studied in mathematical chemistry [14,15,25,29,35]. The Moore-Penrose generalised inverse (or the pseudo-inverse) L + of the graph Laplacian L, which has been proved to exist for any connected graph, gives the following formula [15,25,29] for computing the resistance distance: Let L(i) be the matrix resulting from removing the ith row and column of the Laplacian and let L(i, j ) the matrix resulting from removing both the ith and j th rows and columns of L. Then, it has been proved that the resistance distance can be also calculated as…”

“…Then, the average path lengthl is given byl = 2W(G)/n(n− 1). The analogue of the Wiener index in the context of the resistance distance matrix is known as the Kirchhoff index Kf and is defined as Kf = i<j Ω ij [6,25,35,36,38]. It is known that Kf can be expressed in terms of the Laplacian eigenvalues as follows [35]:…”

“…where ψ µ is the eigenvector of the mode µ and L + is again the Moore-Penrose generalised inverse of the graph Laplacian [35]. The case i = j gives the mean square displacement (∆x i ) 2 ≡ x 2 i in (2.11).…”

Unveiling community structures in weighted networks

“…To describe the overall behavior of a walker on G, one needs to go beyond the usual analysis of Markov chains with transition matrix P ij , probability to go from vertex i to an adjacent vertex j, to include also hops, i.e., moves across the graph. For this end, we evaluate the effective resistances r ij between all distinct vertices i and j of G. Those effective resistances r ij can be numerically evaluated by means of the electrical network theory as [27,28] …”